Systems of Equations Introduction

Grade 8 · Math · Study Guide

Learn how to solve two equations with two unknowns using substitution and graphing methods.

A system of equations is a set of two or more equations that share the same variables. The solution to a system is the point (x, y) that makes both equations true at the same time.

There are three main ways to solve a system of linear equations. The first is graphing: you graph both lines on the same coordinate plane and find where they intersect. The intersection point is the solution. The second is substitution: you solve one equation for one variable, then substitute that expression into the other equation. The third is elimination: you add or subtract the equations to eliminate one variable.

Systems can have exactly one solution (the lines intersect at one point), no solution (the lines are parallel and never meet), or infinitely many solutions (the lines are the same line, called coincident lines).

For example, to solve the system y = 2x + 1 and y = -x + 4 by substitution, set the two right sides equal: 2x + 1 = -x + 4. Solving gives x = 1, then y = 3. The solution is (1, 3).

Key terms

System of equations: A set of two or more equations with the same variables
Solution: An ordered pair (x, y) that satisfies all equations in the system
Substitution method: Solving one equation for a variable and plugging it into the other equation
Elimination method: Adding or subtracting equations to remove one variable
No solution: When two lines are parallel and never intersect
Infinitely many solutions: When two equations describe the same line

Flashcards

System of equations: Two or more equations with the same variables solved together
One solution: Lines intersect at exactly one point; the most common case
No solution: Lines are parallel; same slope, different y-intercepts
Infinite solutions: Lines are identical; same slope and same y-intercept
Substitution: Isolate one variable, substitute into the other equation
Elimination: Add or subtract equations to cancel out one variable
Intersection point: The (x, y) coordinate where two lines cross; the solution
Consistent system: A system that has at least one solution

Questions

What is the solution to the system y = x + 2 and y = 3x - 4?
1. (3, 5)
2. (2, 4)
3. (1, 3)
4. (4, 8)
Show answer
Answer: (3, 5)
Set x + 2 = 3x - 4, so 6 = 2x, x = 3. Then y = 3 + 2 = 5.
A system of two linear equations can have exactly two solutions.
Show answer
Answer: False
Two lines can intersect at most once, so a system has 0, 1, or infinitely many solutions — never exactly two.
When two lines in a system are parallel, the system has _____ solution(s).
Show answer
Answer: no
Parallel lines never intersect, so there is no point satisfying both equations.
Which method involves solving one equation for a variable and plugging it into the other?
1. Graphing
2. Elimination
3. Substitution
4. Factoring
Show answer
Answer: Substitution
Substitution means solving for one variable and plugging that expression into the other equation.
The point (2, 5) is a solution to both y = 2x + 1 and y = x + 3.
Show answer
Answer: True
Check: 5 = 2(2)+1 = 5 ✓ and 5 = 2+3 = 5 ✓. Both equations are satisfied.

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